Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $x = \dfrac{8q}{10(4q + 7)} \div \dfrac{2}{36q + 63} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{8q}{10(4q + 7)} \times \dfrac{36q + 63}{2} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 8q \times (36q + 63) } { 10(4q + 7) \times 2 } $ $ x = \dfrac {8q \times 9(4q + 7)} {2 \times 10(4q + 7)} $ $ x = \dfrac{72q(4q + 7)}{20(4q + 7)} $ We can cancel the $4q + 7$ so long as $4q + 7 \neq 0$ Therefore $q \neq -\dfrac{7}{4}$ $x = \dfrac{72q \cancel{(4q + 7})}{20 \cancel{(4q + 7)}} = \dfrac{72q}{20} = \dfrac{18q}{5} $